PSI - Issue 52

Jan Sladek et al. / Procedia Structural Integrity 52 (2024) 133–142 Author name / Structural Integrity Procedia 00 (2019) 000–000

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not sufficiently fast. Time needed to create arrest-holes in cracked structures can be too long. However, the thermal effects on the stress distribution around the crack tip in electrically conducting materials caused by the electric current (Cai and Yuan, 1999) can be very promising due to a fast repair action. Locally higher values of the electric current density are occurring at the crack tip vicinity. The Joule heating induces high temperature gradients around the crack front. Consequently, also additional thermal stresses and deformations are induced in cracked bodies. Song et al. (2019) investigated thermal stresses induced around the interface crack tip by the electric current in macro-sized structures. At the crack tip vicinity, there are large strain gradients, which should be considered in a more advanced continuum model than in the classical mechanics. In micro/nano-sized structures, the application of the classical continuum mechanics cannot be applied due to the size effect. Nonlocal or gradient theories are able to capture the size effect in structures for mechanical fields (Aifantis 2003, Eringen 1976). However, there observed also large temperature gradients at the crack tip vicinity. For better understanding the heat transport in such a case, it is needed to apply a model, where these temperature gradients are considered (Allen 2014, Sladek et al. 2021). In the present paper, the Joule heating as a multi-physical problem is modelled by the gradient theory for both thermal and mechanical fields. It is the first effort to consider the size effects for the mechanical and thermal transport equations. The order of partial differential equations in gradient theories is higher than in their classical counterpart. Therefore, the standard C 0 continuous approximations in the finite element method (FEM) cannot be applied. The collocation mixed FEM (Tian et al. 2021) is applied to our multiphysical problem, where the C 0 continuous approximations are applied independently to displacements and strains and also to temperature and temperature gradients. The collocation method is used at internal points of elements to satisfy kinematic constraints between these quantities.

Nomenclature Q

Joule heating

flux of electric current electric intensity vector electric conductivity electric potential thermal conductivity heat flux vector electric current

H j E i J

ij s

 ij   

i 

temperature mass density specific heat

c

ik m

higher grade flux

T l

internal length thermal material parameter normal derivative of temperature

p P

high order heat flux linear thermal expansion

kl 

Cauchy stress

ij 

higher order stress

jkl 

strains

ij 

klmn c

elastic coefficients

internal length elastic material parameter

l

i t i s i u

traction vector

normal derivative of displacement

displacements